The Poisson distribution and my blisters

How I used the Poisson distribution to see how likely I am to get a minor injury when running.

What is the the Poisson distribution?

From iSixSigma

Discrete distribution.

Length of the observation period (or area) is fixed in advance.

Events occurs at a constant average rate.

Occurrences are independent.

Rare event.

Uses include:

Number of events in an interval of time (or area) when the events are occurring at a constant rate.

Number of items in a batch of random size.

Design reliability tests where the failure rate is considered to be constant as a function of usage.

The Poisson distribution is concerned with the number of randomly occurring events per unit of time or space. It was originally derived as a limiting form of the binomial distribution in the sense that it acts as a good approximation for calculating binomial probabilities when the number of trials n is very large (and tends towards infinity) and when the probability of success at each trial, p, is very small (and tends towards zero). Calculation of probabilities in such situations using the binomial formula is extremely tedious. The use of the Poisson distribution, however, greatly facilitates computation as we shall see below.

The Poisson distribution is important in its own right, however, as a ‘model’ for random events and has a wide range of applications. Its main use is in problems dealing with rare events which occur independently and randomly in continuous situations such as space, length or time.

Examples of Usage:

A classic example, noted by the discoverer of the distribution in the last century, was the number of Prussian soldiers kicked to death by horses in the course of a year. These deaths were found to follow what is known as a Poisson probability distribution. Nowadays, it is found that many applications arise in situations involving:

waiting time or queuing such as the number of customers passing through a supermarket checkout per hour
the number of cars arriving at a car park per day
the number of cash withdrawals from a cash dispenser per hour, etc.

These problems involve a variable generated over time.

Other situations involve space (defined by length, area or volume) such as:

the number of leaks along an oil pipeline
the number of defects in the production of a particular area of plastic sheeting
the number of fish in a given volume of ocean, etc.

The attention in all these situations is on the occurrence of relatively rare events because from a practical viewpoint it is easier to observe the (relatively small) number of times such events occur than it is to observe the number of times such events do not occur.

My blisters
I keep a running journal and I found that the average number of minor injuries I sustained while running over a typical one month period is five. My feet get hammered: Each one pounds the ground some 800 times per mile with three times my body-weight. So its no wonder that I get things like blisters, muscle soreness, etc.

I wanted to find out the probability that on any particular month I’ll get exactly five minor injuries.

The Poisson distribution is appropriate here because: